Key to the development of biological systems, from simple swarming amoeboid protozoans, to human embryos, is patterning. Signaling networks within cells can interact with each other across cell membranes to produce patterns as emergent properties of Iocal interactions. Experimental biologists have developed an impressive array of tools to probe developmental signaling mechanisms, however, the complexity of even relatively simple signaling networks dictates that the relationship between a specific molecular or genetic intervention and the resulting tissue level pattern will defy intuitive understanding. The goal of this proposed research is to design a modeling framework, hierarchical hybrid systems, and accompanying mathematical analysis and computational tools, to interrogate the properties of proposed signaling networks which cause patterning. Hybrid systems evolve continuously, as well as have a phased operation which can represent the logical switching on and off of different mechanisms, control laws, or governing dynamics. Hybrid system models have analytic and computational advantages over continuous models, because they can be used to represent large complex systems as systems of smaller, interacting continuous subsystems, which are easier to analyze. The testbed for this study will be a mechanism called planar cell polarity (PCP) which polarizes epithelial cells within the plane of the membrane. In Drosophila, PCP interacts with other mechanisms which produce hairs on the wing, and segmentation and joint formation in the legs: mutations in PCP induce distorted hair and joint patterns. In humans, disruptions in PCP appear to result in congenital deafness syndromes and neural tube closure defects leading to spina bifida and related conditions. The aims of the proposed research are thus to (I) develop mathematical modeling and analysis tools for hierarchical hybrid systems, and (2) apply these to problems in PCP in Drosophila. The long term goals are to use this research to eventually aid in understanding the corresponding signaling in humans.